1. 九大数理集中講義 Comparison, Analysis, and Control of Biological Networks (3) Domain-Based Mathematical Models for Protein Evolution Tatsuya Akutsu Bioinformatics Center Institute for Chemical Research Kyoto University

2. Contents A simple evolutionary model of protein domains A domain-based model of protein-protein interaction networks An evolutionary model of multi domain proteins

3. Motivation of Our Studies Explaining observed distributions on proteins and PPI networks  PPI networks are scale-free [Jeong et al., 2001]  #(proteins having k domain families) follows exponential distribution [Koonin et al., 2002]  #(proteins having k domains) follows power-law [Koonin et al., 2002]  #(domains appearing in k proteins) follows power-law [Wuchty, 2001] Providing simple evolutionary models  In real proteins, what evolve are not networks but genes/proteins

4. An Evolutionary Model of Protein Domains J.C. Nacher, M. Hayashida and T. Akutsu: Physica A, 367, 538-552, 2006

5. Protein Domain Domain: Well-defined region within a protein that either performs a specific function or constitutes a stable unit Protein consisting of 3 domains

6. Evolutionary Model of Protein Domains N proteins, each one consists of only one domain (domains are different from each other) We repeat T times the following steps: a)With probability (1-a) we create a new protein with new domain (MUTATION) b) Otherwise, we randomly select one protein and make a copy of it (PROTEIN DUPLICATION) We assume that each protein consists of only one domain

7. Model (continued) i : i -th kind of domain : number of proteins consisting of i -th domain : time when i -th domain was first created 1-a a Duplication of Protein T times a ~ 1.0 Mutation Q(k): number of domains each of which appears in k proteins As in Barabasi & Albert 1999

8. Protein duplication mutation Prob.= a Prob.= 1- a Model of Protein Evolution

9. Exaplanation of Q(k) 12345 Types of domains 6 Types of proteins

10. Our Model vs. Preferential Attachment Similarity  #(proteins with the i-th domain) ⇔ degree of the i-th node  Duplication of protein with the i-th domain ⇔ Attachment of an edge to the i-th node  Mutation (creation of a protein with a new domain) ⇔ Addition of a new vertex Difference: new node new edge 1-a a Duplication a ~ 1.0 Mutation P D (1)=3 P D (2)=1 P D (3)=1

11. A Domain-Based Model of Protein- Protein Interaction Networks J.C. Nacher, M. Hayashida and T. Akutsu: BioSystems, 95, 155-159, 2009

12. A Domain-Based Model of Protein-Protein Interactions [Sprinzak & Margalit 2001, Deng et al. 2002] Proteins interact ⇔ There exist interacting domain pair(s) AB CD X Y Z Domain-Domain Interaction Protein-Protein Interaction

13. Combination of Domain Evolution Model and Domain-based Protein-Protein Interaction Model Evolutional model of protein domains Random interaction of domains Domain-based protein-protein interactions Proteins interact ⇔ There exist interacting domain pair(s) Scale-free property of PPI (protein-protein interaction network)

14. Mathematical Analysis domain A domain B n A =x =3 n B =y =2 3 proteins with degree 2 However, if the number of domain-domain interactions is large, the distribution approaches to the normal distribution because of the central limit theorem

15. An Evolutionary Model of Multi Domain Proteins J.C. Nacher, M. Hayashida and T. Akutsu: BioSystems, 101:127-135, 2010.

16. Domain Fusion and Internal Duplication (1) 1. Internal Duplication  Duplication of one or more domains inside one protein 2. Domain Fusion  Two proteins are merged Domain Fusion Mutation Protein Duplication Internal Domain duplication

17. Modeling of Duplication, Mutation and Fusion (1) N i (t) : #proteins having i domains at time t p m : prob. mutation (creation of new protein) occurs p d : prob. duplication occurs p f : prob. fusion occurs

18. Modeling of Duplication, Mutation and Fusion (2) By letting n i (t) =N i (t) /t and n i = n i (t) for t→∞

19. Modeling of Duplication, Mutation and Fusion (3) Using generation function, we have exact solution Using Stirling’s approximation It shows n k follows almost exponential distribution

20. Modeling of Internal Duplication By letting n i (t) =N i (t) /t and n i = n i (t) for t→∞ n k follows power-law

21. Combination of Mutation, Fusion, Internal/External Duplications Difficult to solve ⇒ Computer simulation

22. Summary A simple (simplest? ) model of protein domain evolution, which explains power-raw distribution A domain-based model of protein-protein interaction network ⇒ Explains power-law property of PPI ⇒ Good agreement between simulation and real data ⇒ Simpler than existing models (e.g., duplication-divergence) An evolutionary model of multi-domain proteins ⇒ #(proteins having k domain families) follows exponential ⇒ #(proteins having k domains) follows power-law ⇒ Good agreement between simulation and real data ⇒ Importance of role of internal duplications